Disordered topological insulators via C ^* -algebras

Abstract

Abstract. -The theory of almost commuting matrices can be used to quantify topological obstructions to the existence of localized Wannier functions with time-reversal symmetry in systems with time-reversal symmetry and strong spin-orbit coupling. We present a numerical procedure that calculates a Z2 invariant using these techniques, and apply it to a model of HgTe. This numerical procedure allows us to access sizes significantly larger than procedures based on studying twisted boundary conditions. Our numerical results indicate the existence of a metallic phase in the presence of scattering between up and down spin components, while there is a sharp transition when the system decouples into two copies of the quantum Hall effect. In addition to the Z2 invariant calculation in the case when up and down components are coupled, we also present a simple method of evaluating the integer invariant in the quantum Hall case where they are decoupled.The study of topological insulators is one of the most active areas of physics today. Experimental and theoretical work has shown physical realizations of time-reversal invariant insulators with strong spin-orbit coupling in both two [1] and three dimensions [2] and a complete classification of different insulating phases has been recently obtained using methods of Anderson localization [3] and, more generally, K-theoretic techniques [4].However, numerically it is difficult to determine the Z 2 invariants that are signatures of topological insulating phases. For systems with translational invariance, one can study the bundle over the momentum torus [5], while for systems without translation invariance, Essin and Moore [6] were able to study the phase diagram of a graphene model by studying the model over a flux torus corresponding to twisted boundary conditions. Unfortunately, the flux torus approach is very computationally intensive: for each disorder realization, the Hamiltonian must be diagonalized once for each point on a discrete grid on the flux torus, and then the connection on the torus must be computed. This limited the study to small systems, with at most 64 sites.In this paper, we present a different approach to calculating a Z 2 invariant, based on ideas in C * -algebras, in particular the K-theory of almost commuting matrices. We present a fast numerical algorithm based on these ideas. Computing the invariant requires a single diagonalization of the Hamiltonian, matrix function calculations on matrices at most half the size of the Hamiltonian, and finally the calculation of the Pfaffian of a real anti-symmetric matrix that is at most the size of the Hamiltonian. The most costly step is a single diagonalization, allowing us to study significantly larger samples, up to 1600 sites.Our invariant serves the same fundamental purpose as the invariant used in [6]-proving that for certain lowenergy bands it is impossible to find well-localized Wannier functions with time-reversal symmetry. These invariants are most likely equivalent, but that is another topic [7].We apply ...

Disordered topological metals

“…As noted by several authors, 22,23,[28][29][30][31] disorder produces a phase transition between the topological insulator phase and a gapless metallic phase. This can be seen most directly using the self-consistent Green function, given by Eqs.…”

“…Upon averaging, one may, thus, obtain a nonquantized value. It has already been pointed out by Loring and Hastings,22,23 …”

“…17,18 Theoretical studies of doping showed that the metal obtained when the chemical potential of the TI lies outside the band gap is characterized by a finite, but nonquantized topological index. [19][20][21][22][23] Studies of disorder led to additional surprises: It was shown that disorder can induce topological behavior in some trivial insulators, [24][25][26][27] and (more importantly for the current work) that disorder may close the gap in topological insulators, leading to a metallic phase. [28][29][30][31][32] In our work, we focus on the disorder-induced metallic phase in a simple test case of the Kane-Mele-Haldane honeycomb model.…”

Pumping conductance, the intrinsic anomalous Hall effect, and statistics of topological invariants

“…To study larger system sizes and obtain a reasonable test of the conjecture regarding Chern numbers, we use the method of Hastings and Loring 3,34 . Let P again be the projection operator onto some set of bands.…”

“…This phase shows up as a region of fillings over which the Chern number continues to fluctuate as the system becomes larger. 3,21 We will focus in this paper on approaching non-interacting metals in the unitary symmetry class (i.e., with broken timereversal symmetry), essentially by scaling disorder to zero as system size grows. The details of this approach are provided in Section III.…”